Maple Questions and Posts

How to verify the solution of PDEs?...

Asked by: Ahmed111 130

May 21 2020

1 1

>

restart:

>	with(PDEtools):

>	with(LinearAlgebra):

>

>	alias(f=f(x,t),g=g(x,t));

(1)

>

>	eq1:=diff(f,x)=-Ietaf +Iexp(-It)*g;

(2)

>	eq2:=diff(g,x)=-Ietag +Iexp(It)*f;

(3)

>	eq3:=diff(f,t)=(Ieta^2-I/2)f +Ietaexp(-It)g;

(4)

>	eq4:=diff(g,t)=(-Ieta^2+I/2)g +Ietaexp(It)f;

(5)

>	#### The solution of (2)-(5) is

>	eq5:=f=I(c1exp(A)-c2exp(-A))exp(-i*t/2);

(6)

>	eq6:=g=(c2exp(A)-c1exp(-A))exp(it/2);

(7)

>	#### where

>	c1=sqrt(h-sqrt(h^2-1))/sqrt(h^2-1);c2=sqrt(h+sqrt(h^2-1))/sqrt(h^2-1);A=sqrt(h^2-1)(x+Ih*t);

c1 = (h-(h^2-1)^(1/2))^(1/2)/(h^2-1)^(1/2)

c2 = (h+(h^2-1)^(1/2))^(1/2)/(h^2-1)^(1/2)

(8)

>	#### How to verify (6) and (7) is the solution of (2)-(5)?

Download verification.mw

The equations of motion in curvilinear coordinates,...

by: ecterrab 13431 Product: Maple

May 21 2020

5 0

The equations of motion in curvilinear coordinates, tensor notation and Coriolis force

The formulation of the equations of motion of a particle is simple in Cartesian coordinates using vector notation. However, depending on the problem, for example when describing the motion of a particle as seen from a non-inertial system of references (e.g. a rotating planet, like earth), there is advantage in using curvilinear coordinates and also tensor notation. When the particle's movement is observed from such a rotating referential, we also see "acceleration" that is not due to any force but to the fact that the referential itself is accelerated. The material below discusses and formulates these topics, and derives the expression for the Coriolis and centripetal force in cylindrical coordinates as seen from a rotating system of references.

The computation below is reproducible in Maple 2020 using the Maplesoft Physics Updates v.681 or newer.

Vector notation

Generally speaking the equations of motion of a particle are easy to formulate: the position vector is a function of time, the velocity is its first derivative and the acceleration is its second derivative. For instance, in Cartesian coordinates

>

(1)

>

(2)

>

(3)

Newton's 2^nd law, that in an inertial system of references when there is force there is acceleration and viceversa, is then given by

>

(4)

where represents the total force acting on the particle. This vectorial equation represents three second order differential equations which, for given initial conditions, can be integrated to arrive at a closed form expression for as a function of .

Tensor notation

In Cartesian coordinates, the tensorial form of the equations (4) is also straightforward. In a flat spacetime - Galilean system of references - the three space coordinates form a 3D tensor, and so does its first derivate and the second one. Set the spacetime to be 3-dimensional and Euclidean and use lowercaselatin indices for the corresponding tensors

>

Physics:-g_[mu, nu] = Matrix(%id = 18446744078329083054)

(5)

The position, velocity and acceleration vectors are expressed in tensor notation as done in (1), (2) and (3)

>

(6)

>

(7)

>

(8)

Setting a tensor to represent the three Cartesian components of the force

>

${Physics:-Dgamma[a], F[j], Physics:-Psigma[a], Physics:-d_[a], Physics:-g_[a, b], Physics:-LeviCivita[a, b, c], Physics:-SpaceTimeVector[a](X)}$

(9)

Newton's 2^nd law (4), now expressed in tensorial notation, is given by

>

(10)

The three differential equations behind this tensorial form of (4) are as expected

>

${F__x(t) = m*(diff(diff(x(t), t), t)), F__y(t) = m*(diff(diff(y(t), t), t)), F__z(t) = m*(diff(diff(z(t), t), t))}$

(11)

Things are straightforward in Cartesian coordinates because the components of the line element are exactly the components of the tensor

>

(12)

and so, the form factors (see related Mapleprimes post) are all equal to 1.

In the case of no external forces, and the equations of motion, whose solution are the trajectory, can be formulated as the path of minimal length between two points, a geodesic. In the case under consideration, because the spacetime is flat (Galilean) those two points lie on a plane, we are talking about Euclidean geometry, that information is encoded in the metric (the 3x3 identity matrix (5)), and the geodesic is a straight line. The differential equations of this geodesic are thus the equations of motion (11) with , and can be computed using Geodesics

>

(13)

Curvilinear coordinates

Vector notation

The form of these equations in the case of curvilinear coordinates, for example in cylindrical or spherical variables, is obtained performing a change of coordinates.

>

(14)

This change keeps the z axis unchanged, so the corresponding unit vector remains unchanged.

Changing the basis and coordinates used to represent the position vector , it becomes

>

(15)

where since in (1) the coordinates () depend on t, in (15), not just and but also the unit vector depends on t. The velocity is computed as usual, differentiating

>

(16)

The second term is due to the dependency of on the coordinate together with the chain rule diff(`#mover(mi("ρ",fontstyle = "normal"),mo("&and;"))`(t), t) = (diff(`#mover(mi("ρ",fontstyle = "normal"),mo("&and;"))`, phi))*(diff(phi(t), t)) and (diff(`#mover(mi("ρ",fontstyle = "normal"),mo("&and;"))`, phi))*(diff(phi(t), t)) = `#mover(mi("φ",fontstyle = "normal"),mo("&and;"))`(t)*(diff(phi(t), t)) . The dependency of curvilinear unit vectors on the coordinates is automatically taken into account when differentiating due to the Setup setting geometricdifferentiation = true.

For the acceleration,

>

(17)

where the term involving comes from differentiating in (16) taking into account the dependency of on the coordinate This result can also be obtained by directly changing variables in the acceleration , in equation (3)

>

(18)

Newton's 2^nd law becomes

>

(19)

In the absence of external forces, equating to 0 the vector components (coefficients of the unit vectors) of the acceleration we get the system of differential equations in cylindrical coordinates whose solution is the trajectory of the particle expressed in the (

>

$`~`[`=`]({coeffs(rhs(diff(diff(r_(t), t), t) = _rho(t)*(diff(diff(rho(t), t), t)-rho(t)*(diff(phi(t), t))^2)+_phi(t)*(2*(diff(rho(t), t))*(diff(phi(t), t))+rho(t)*(diff(diff(phi(t), t), t)))+(diff(diff(z(t), t), t))*_k), [`#mover(mi("ρ",fontstyle = "normal"),mo("&and;"))`(t), `#mover(mi("φ",fontstyle = "normal"),mo("&and;"))`(t), `#mover(mi("k"),mo("&and;"))`])}, 0)$

${2*(diff(rho(t), t))*(diff(phi(t), t))+rho(t)*(diff(diff(phi(t), t), t)) = 0, diff(diff(rho(t), t), t)-rho(t)*(diff(phi(t), t))^2 = 0, diff(diff(z(t), t), t) = 0}$

(20)

>	$solve({2(diff(rho(t), t))(diff(phi(t), t))+rho(t)(diff(diff(phi(t), t), t)) = 0, diff(diff(rho(t), t), t)-rho(t)(diff(phi(t), t))^2 = 0, diff(diff(z(t), t), t) = 0}, {diff(phi(t), t, t), diff(rho(t), t, t), diff(z(t), t, t)})$

${diff(diff(phi(t), t), t) = -2*(diff(rho(t), t))*(diff(phi(t), t))/rho(t), diff(diff(rho(t), t), t) = rho(t)*(diff(phi(t), t))^2, diff(diff(z(t), t), t) = 0}$

(21)

In this formulation (21) with , although , no acceleration in the direction, is naturally expected, the same cannot be said about the other two equations for and . Those two equations are discussed below under Coriolis and Centripetal forces. The key observation at this point, however, is that the right-hand sides of both unexpected equations involve , rotation around the z axis.

Tensor notation

The same equations (19) and (21) result when using tensor notation. For that purpose, one can transform the position, velocity and acceleration tensors (6), (7), (8), but since they are expressed as functions of a parameter (the time), it is simpler to transform only the underlying metric using TransformCoordinates. That has the advantage that all the geometrical subtleties of curvilinear coordinates, like scale factors and dependency of unit vectors on curvilinear coordinates, get automatically, very succinctly, encoded in the metric:

>

Physics:-g_[a, b] = Matrix(%id = 18446744078263848958)

(22)

The computation of geodesics assumes that the coordinates () depend on a parameter. That parameter is passed as the first argument to Geodesics

>

(23)

These equations of motion (23) are the same as the equations (21) computed using standard vector notation, differentiating and taking into account the dependency of curvilinear unit vectors on the curvilinear coordinates in (16) and (17). One of the interesting features of computing with tensors is, as said, that all those geometrical algebraic subtleties of curvilinear coordinates are automatically encoded in the metric (22).

To understand how are the geodesic equations computed in one go in (23), one can perform the calculation in steps:

1.	Make be a function of directly in the metric

2.	Compute - not the final form of the equations (23) - but the intermediate form expressing the geodesic equation using tensor notation, in terms of Christoffel symbols

3.	Compute the components of that tensorial equation for the geodesic (using TensorArray)

For step 1, we have

>

Physics:-g_[a, b] = Matrix(%id = 18446744078354237910)

(24)

Set this metric where

>

Physics:-g_[a, b] = Matrix(%id = 18446744078342604430)

(25)

Step 2, the geodesic equations in tensor notation with the coordinates depending on the time t are computed passing the optional argument tensornotation

>

(26)

Step 3: compute the components of this tensorial equation

>

(27)

These are the same equations (23).

Having the tensorial equation (26) is also useful to formulate the equations of motion in tensorial form in the presence of force. For that purpose, redefine the contravariant tensor to represent the force in the cylindrical basis

>

${Physics:-D_[a], Physics:-Dgamma[a], F[j], Physics:-Psigma[a], Physics:-Ricci[a, b], Physics:-Riemann[a, b, c, d], Physics:-Weyl[a, b, c, d], Physics:-d_[a], Physics:-g_[a, b], Physics:-Christoffel[a, b, c], Physics:-Einstein[a, b], Physics:-LeviCivita[a, b, c], Physics:-SpaceTimeVector[a](X)}$

(28)

Newton's 2^nd law (19)

>

(29)

now using tensorial notation, becomes

>

(30)

>

Array(%id = 18446744078329063774)

(31)

where we recall (see related Mapleprimes post) that to obtain the vector components entering from these tensor components we need to multiply the latter by the scale factors (), the component of in the direction of is given by .

Coriolis force and centripetal force

After changing variables the position vector of the particle got expressed in (15) as

A distinction needs to be made here, according to whether the unit vector depends or not on the time , the former being the general case. When is a constant, the value of the coordinate - the angle between and the x axis - does not change, there is no rotation around the z axis. On the other hand, when , the orientation of and so the coordinate changes with time, so either the force acting on the particle has a component in the direction that produces rotation around the z axis, or the system of references - itself - is rotating around the z axis.

Likewise, the expression (15) can represent the position vector measured in the original Galilean (inertial) system of references, where a force is producing rotation around the z axis, or it can represent the position of the particle measured in a rotating, non-inertial system references. Hence the transformation (14) can also be interpreted in two different ways, as representing a choice of different functions (generalized coordinates) to represent the position of the particle in the original inertial system of references, or it can represent a transformation from an inertial to another rotating, non-inertial, system of references.

This equivalence between the trajectory of a particle subject to an external force, as observed in an inertial system of references, and the same trajectory observed from a non-inertial accelerated system of references where there is no external force, actually at the root of the formulation of general relativity, is also well known in classical mechanics. The (apparent) forces observed in the rotating non-inertial system of references, due only to its acceleration, are called Coriolis and centripetal forces.

To see that the equations

diff(rho(t), t, t) = (diff(phi(t), t))^2*rho(t), diff(phi(t), t, t) = -2*(diff(phi(t), t))*(diff(rho(t), t))/rho(t)

that appeared in (27) when in the inertial system of references , are related to the Coriolis and centripetal forces in the non-inertial referencial, following [1] introduce a vector representing the rotation of that referencial around the z axis (when, in the inertial system of references, the non-inertial system rotates clockwise, in the non-inertial system increases in value in the anti-clockwise direction)

>

(32)

According to [1], (39.7), the acceleration in the inertial system is expressed in terms of the quantities in the non-inertial rotating system by the sum of the following three vectorial terms.

First, the components of the acceleration measured in the non-inertial system are given by the second derivatives of the coordinates () multiplied by the scale factors, which in this case are given by () (see this post in Mapleprimes)

>

(33)

Second, the Coriolis force divided by the mass, by definition given by

>

(34)

Third the centripetal force divided by the mass, defined by

>

(35)

Adding these three terms,

>

a_(t)+2*Physics:-Vectors:-`&x`(diff(r_(t), t), omega_)+Physics:-Vectors:-`&x`(omega_, Physics:-Vectors:-`&x`(r_(t), omega_)) = _rho(t)*(diff(diff(rho(t), t), t)-rho(t)*(diff(phi(t), t))^2)+_phi(t)*(2*(diff(rho(t), t))*(diff(phi(t), t))+rho(t)*(diff(diff(phi(t), t), t)))+(diff(diff(z(t), t), t))*_k

(36)

So that

>

(37)

and where the right-hand side of (36) is, actually, the result computed lines above in (18)

>

(38)

>

(39)

From (37), in the absence of external forces and so the acceleration measured in the rotating system is given by the sum of the Coriolis and centripetal accelerations

>

(40)

In other words: in the absence of external forces, the acceleration of a particle observed in a rotating (non-inertial) system of references is not zero.

Expressing this equation (40) in terms of () we get

>

(41)

resulting in the three equations

>

(42)

>

(43)

>

(44)

which are the equations returned by Geodesics in (23)

>

(45)

>

References

[1] L.D. Landau, E.M. Lifchitz, Mechanics, Course of Theoretical Physics, Volume 1, third edition, Elsevier.

Download The_equations_of_motion_in_curvilinear_coordinates.mw

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

Atomic Variables ...

Asked by: dareimert 35

May 21 2020

0 1

I'm trying to document some generator calculations that use terms line Xd'. In Maple that turns into a derivative. I converted several 'ed variables into atomic variables. They have made my work sheet unusable. They appear to randomly loose connecin to their assigned values and prevent numeric solutions by showing up in symbolic form. Once this happens, I have to check each instance where the variable isuse to find where the loss/lossses occured. (copy and past each instance of the variable and check its value).

Once found I can't find a consistant way to re-associate them with the assigned value.

Any suggestions. See botton of attaced work sheet.

Maple Worksheet - Error

Failed to load the worksheet /maplenet/convert/WS_Gen_I_and_E_for_fauls_on_HS_of_GSU_-_Time_Response_&_51V_Trip_Times_-R5a.mw .

Download WS_Gen_I_and_E_for_fauls_on_HS_of_GSU_-_Time_Response_&_51V_Trip_Times_-R5a.mw

Maple Worksheet - Error

Failed to load the worksheet /maplenet/convert/WS_Gen_I_and_E_for_fauls_on_HS_of_GSU_-_Time_Response_&_51V_Trip_Times_-R5a.mw .

Download WS_Gen_I_and_E_for_fauls_on_HS_of_GSU_-_Time_Response_&_51V_Trip_Times_-R5a.mw

How to define 4 functions from the dsolve solution...

Asked by: TomVXP 15

May 21 2020

1 3

How to define 4 functions from the dsolve solution of a system of 4 differential equations?

restart: with(plots): with(DEtools):with(LinearAlgebra):with(Statistics):with(CurveFitting):with(Optimization):
ddesys := {diff(S(t),t) = -beta*S(t)*Ix(t)/N,
diff(Ex(t),t) = beta*S(t)*Ix(t)/N - sigma*Ex(t-tau__1),
diff(Ix(t), t) = sigma*Ex(t - tau__1)- gamma*Ix(t-tau__2),
diff(R(t),t) = gamma*Ix(t-tau__2),
diff(Dx(t),t) = delta*Ix(t),
S(0) = 80900, Ex(0) = 1, Ix(0) = 1, R(0) = 0, Dx(0) = 0 }:
dsn := dsolve(eval(ddesys, {beta = 4, gamma = 0.0478, sigma = 0.10, delta = 0.0005, N=80900, tau__1 = 1.1,tau__2 = 8.7,tau__3 = 0}), numeric):

How to solve a stochastic partial differential equ...

Asked by: Dmitrii 15

May 21 2020

1 2

Hi,

Please can you give me a hand with numerical solving and visualising the partial differential wave equation with stochastic term eta(t), using methods of stochastic calculus?

diff(u(x, t), t $ 2) - (1+eta(t))*diff(u(x, t), x $ 2) = 0

I had a look at the "stochastic" package by Sasha Cyganowski, but couldn't find an example for stochastic pde.

Look forward to your help.

Thanks,

Dmitrii

A probability riddle...

Asked by: vv 12453

May 21 2020

2 16

Here is a known probability riddle:

A and B are two lists of 100 binary numbers:

A:=[0,1,0,1,0,1,1,0,1,1,1,0,0,1,1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0,1,0,1,0,1,0,1,1,1,1,1,0,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0,1,1,1,1,0,0,0,1,0,1,0,0,0,0,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,1,0,1,0]:
B:=[0,1,1,0,1,1,0,1,1,1,0,1,0,0,1,1,0,0,0,0,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,0,1,0,1,1,0,1,1,0,0,0,0,0,0,1,0,1,1,1,0,0,0,1,0,1,0,0,0,0,1,0,0,0,0,1,0,1,1,1,1,1,1,0,0,0,1,1,1,1,1,1,1,0,0,1,0]:

One was obtained by tossing a coin (1 for a head, 0 for a tail), and the other by a human, who was asked to simulate tossing a coin.

Question: which one comes from a human brain?
The standard answer: B was produced using a coin, because (among other things) the probabilty of obtaining a "000000" or "111111" is about 80%, but a humain brain tends to avoid such "simulations".

My Question: what (if any) statistical test can be used in Maple for an answer?
(I have tried ChiSquareSuitableModelTest but both lists were accepted).

I have a set of boundary conditions that i want t...

Asked by: Ridhaaa 5

May 20 2020

1 0

# I solved the differential equation using a fourier series decomposting method. I found with the help of "Dsolve" the different expressions of the fourier constants and now I will apply the boundary conditions to find the expressions of its constants.

#this my boundary conditions

#this my system of equations

Why does Maple freeze?...

Asked by: Patrice Sawyer 10

May 20 2020

2 3

Recently Maple started freezing on me for a few minutes; after that it continues to work and then freezes again and so on (it does not so much freezes as it pauses). I do not know what changed on my system (which is Windows 10) for this to happen. I had been using Maple 2019 when this started and then upgraded to Maple 2020 but that did not solve my problem.

It does not seem to have to do with the type of computations I am doing.

The solution of the system of symbolic equations...

Asked by: Grigoriy Yashin 20

May 20 2020

0 6

I have problem with usage of solve for the solution of the system of six and eight symbolic equations. I need to get expressions for variables: R__xl, R__xs, R__xsi, R__zl, R__zs, R__zsi, dbeta__l(t), dbeta__s(t). I tried to find solution for 8 equations, but I got the message: Warning, solutions may have been lost. I thought that the system is non-linear and tried to solve it for six variables (R__xl, R__xs, R__zl, R__zs, R__zsi, R__xsi), in this case equations are linear, but Maple ignored the solve command and did not give a solution or error. Could someone help me with this problem?

Quasi-static.mw

Difference between Threads:-Seq and Grid:-Seq - Wh...

Asked by: emendes 395

May 20 2020

1 2

Hello

I am not sure how to choose between Threads:-Seq and Grid:-Seq.

The problem: a procedure, proc1, that calls two other procedures, proc2 and proc3, verifies if a set of parameters fulfills a certain condition. Proc1, proc2 and proc3 are not part of Maple available functions.

Right now I am using Grid:-Seq to return the result of applying proc1 to chunks of a thousand parameters. I wonder whether I could use Thread:-Seq instead. What are the advantages of using one instead of the other? And when do I choose one over the other?

I would appreciate if you could provide simple examples to explain the differences.

Many thanks.

Ed

PS. There is a problem with Grid:-Seq as reported in one of my previous questions.

Maple 2017.3 problems with complex numbers, solve,...

Asked by: RafalAblamowicz 5

May 20 2020

1 1

Hi,

Maple 2017.3 has these problems which hamper my work with it. I shown three examples.

Example 1: Maple cannot recognize automatically that two complex numbers are equal:

[>eq1:=9*exp((1/9)*(5*I)*Pi)-9*exp((1/9)*(2*I)*Pi) = -8*exp((1/9)*(2*I)*Pi)+7*exp((1/9)*(5*I)*Pi)+exp((1/18)*(13*I)*Pi)*sqrt(3);

Here, lhs(eq1) and rhs(eq1) are in fact equal. Yes, evela(simplify(lhs(eq1)-rhs(eq1)) reduces to zero but not when solving equations (see Example 2).

Example 2:

[>sys1:=[_xx[1]+lhs(eq1),_xx[1]+rhs(eq1)];
[>solve(sys1,[_xx[1],_xx[2]]);

yields no answer.

Example 3:

[>alias(omega=RootOf(_Z^2+_Z+1));
[>rt:=(-1+I*sqrt(3))/2;

Maple fails to substitute the alias and recognize that rt=omega.

Any suggestion on a work-around problems 2 and 3 would be helpful.

Thanks,

Rafal Ablamowicz
www.math.tntech.edu/rafal/

How can I set Physics:-Version(678) active?...

Asked by: Wolfgang Gellien 50

May 20 2020

2 1

I have load the new Physics-Version 678, but it is not active. How can I do this?

Here the output:

restart

version()

User Interface: 1455132
         Kernel: 1455132
        Library: 1455132
       MapleIDE: 928330

1455132

interface(version)

Standard Worksheet Interface, Maple 2020.0, Windows 10, March 4

2020 Build ID 1455132

kernelopts(version)

Maple 2020.0, X86 64 WINDOWS, Mar 4 2020, Build ID 1455132

Physics:-Version()

The "Physics Updates" version in the MapleCloud is 678 and is

the same as the version installed in this computer, created

2020, May 20, 10:21 hours, found in the directory

C:\Users\wgellien\maple\toolbox\Physics Updates\lib\

restart

Physics:-Version(678)

Warning, this package updates content shipped in a standard Maple install. Use the 'restart' command to clear your session before using these commands.
Kernel(The "Physics Updates" version "678" is installed but is

not active. The active version of Physics is within the

library C:\Users\wgellien\maple\toolbox/Physics Updates/lib\P\

hysics Updates.maple, created 2020, May 20, 14:46 hours),

[The "Physics Updates" version "678" is installed but is not

active. The active version of Physics is within the library

C:\Users\wgellien\maple\toolbox/Physics Updates/lib\Physics

Updates.maple, created 2020, May 20, 14:46 hours]

With kind regards

Wolfgang Gellien

Can a second plot be combined with a plot generate...

Asked by: TomVXP 15

May 20 2020

1 3

Can a second plot be added to a plot generated by a plots[odeplot] command for the case below?

> dsn := dsolve(eval(ddesys, {beta = 4, gamma = 0.0478, sigma = 0.10,tau__1 = 1.1,tau__2 = 8.7}), numeric):

> plots[odeplot](dsn, [[t, S(t), color = green], [t, Ex(t), color = black], [t, Ix(t), color = blue],[t, R(t), color = red]], 0 .. 100, legend = [ S(t), Ex(t), Ix(t), R(t)], labels = [t,""] );

Bug in Maple 2019 fixed in 2020...

Asked by: awass 296

May 19 2020

1 2

Beware!

In Maple 2019 we get the strange value

cos((Pi/2) -1e-12);
-10
-2.051033808 10

In Maple 2020 we get the correct answer

cos(Pi/2-1e-12);
-13
9.999999999996916397514419 10

That error in Maple 2019 certainly messed up quite a few of my calcualtions!

POSSIBILITIES IN INCLUSIVE EDUCATION

by: Alsu 445 Product: Maple

May 19 2020

4 0

POSSIBILITIES OF USING OF COMPUTER IN MATHEMATICS

AND OTHER APPLICATIONS IN INCLUSIVE EDUCATION

Alsu Gibadullina, math teacher, math teacher

Secondary and high school # 57, Kazan, Russia

In recent years Russia actively promoted and implemented the so-called inclusive education (IE). According to the materials of Alliance of human rights organizations “Save the children”: "Inclusive or included education is the term used to describe the process of teaching children with special needs in General (mass) schools. In the base of inclusive education is the ideology that ensures equal treatment for everybody, but creates special conditions for children with special educational needs. Experience shows that any of the rigid educational system some part of the children is eliminated because the system is not ready to meet the individual needs of these children in education. This ratio is 15 % of the total number of children in schools and so retired children become separated and excluded from the overall system. You need to understand that children do not fail but the system excludes children. Inclusive approaches can support such children in learning and achieving success. Inclusive education seeks to develop a methodology that recognizes that all children have different learning needs tries to develop a more flexible approach to teaching. If teaching and learning will become more effective as a result of the changes that introduces Inclusive Education, all children will win (not only children with special needs)."

There are many examples of schools that have developed their strategy implementation of IE, published many theoretical and practical benefits of inclusion today. All of them have common, immaterial character. There is no description of specific techniques implementing the principles of IO in the teaching of certain disciplines, particularly mathematics. In this paper we propose a methodology that can be successfully used as in “mathematical education for everyone", also for the development of scientific creativity of children at all age levels of the school in any discipline.

According to the author, one of the most effective methodological tools for education is a computer mathematics (SCM, SSM). Despite the fact that the SCM were created for solving problems of higher mathematics, their ability can successfully implement them in the school system. This opinion is confirmed by more than 10–year-old author's experience of using the package Maple in teaching mathematics. At first it was just learning the system and primitive using its. Then author’s interactive demonstrations, e-books, programs of analytical testing were created by the tools packages. The experience of using the system Maple in teaching inevitably led to the necessity to teach children to work with it. At first worked a club who has studied the principles of the package’s work, which eventually turned into a research laboratory for the use of computer technology. Later on its basis there was created the scientific student society (SSS) “GEODROMhic" which operates to this day. The main idea and the ultimate goal of SSS – individual research activities on their interests with the creation of the author electronic scientific journals through the use of computer mathematics Maple. The field of application of the package was very diverse: from mathematics to psychology and cultural phenomena. SSS’s activity is very high: they are constantly and successfully participate in intellectual high-level activities (up to international). Obviously, not every SSS’s member reaches high end result. However, even basic experience in scientific analysis, modeling, intelligent using of the computer teaches the critical thinking skills, evokes interest to new knowledge, allows you to experience their practical value, gives rise to the development of creative abilities. As a result, the research activity improves intellectual culture, self-esteem and confidence, resistance to external negative influence. It should be noted, however, that members of the scientific societies are not largely the so-called "gifted", than ordinary teenagers with different level of intellectual development and mathematical training. With all this especially valuable is that the student is dealing with mathematical signs and mathematical models, which contributes to the development of mathematical thinking.

From 2007 to 2012. our school (№. 57 of Kazan) was the experimental platform of the Republican study SKM (Maple) and other application software in the system of school mathematical education under the scientific management of Professor Yu. G. Ignatyev of Kazan state University (KF(P)U).

Practical adaptation of computer mathematics and other useful information technologies to the educational process of secondary schools passed and continues to work in the following areas:

The creation of a demonstration support of different types of the lessons;
The embedding of computing to the structure of practical trainings;
In the form of additional courses - studying of computer applications through which you can conduct a research of the mathematical model and create animated presentation videos, web-pages, auto-run menu;
Students’ working on individual creative projects:
- construction of computer mathematical models;
- creating author's programs with elements of scientific researches;
- students create interactive computer-based tutorials;
- creation of an electronic library of creative projects;
The participation of students in the annual competitions and scientific conferences for students;
The accumulation and dissemination of new methodological experience.

Traditionally, the training system has the structure: explanation of a new → the solution of tasks→ check, self-test and control → planning of the new unit with using analysis. However, the main task types: 1) elementary, 2) basic, 3) combined, 4) integrated, 5) custom. With the increasing the level of training a number of basic tasks are growing and some integrated tasks become a class of basic. Thus, the library for basic operations is generated. The decision of the educational task occurs on the way of mastering the theoretical knowledge of mathematical modeling: 1) analysis of conditions (and construction drawing), 2) the search for methods of solution, 3) computation, 4) the researching.

To introduce computer mathematics in this training system, you can:

At demonstrations. For example, with Maplе facilities you can create a step-by-step interactive and animated images, which are essentially the exact graphic interpretation of mathematical models.
If we have centralized collective control.
If students have individual self-control.
In the analysis of the conditions of the problem, for the construction and visualization of its model, the study of this model.
In the computations.
In practical training of different forms.
In individual projects with elements of research.

In the learning process with the use of computer mathematics in the school a library of themed demonstrations, tasks of different levels and purpose, programs, analytical testing, research projects is generated. With all this especially valuable is that the student is dealing with mathematical signs and mathematical models. Addiction to them processed in the course of working with them it’s unobtrusive, naturally, organically.

Mathematical modelling (MM) is increasingly becoming an important component of scientific research. Today's powerful engineering tools allow to carry out numerous computer experiments, deep and full enough of exploring the object, without significant cost painless. Thus provided the advantages of theoretical approach, and experiment. The integration of information technology and MM method is effective, safe and economical. This explains its wide distribution and makes unavoidable component of scientific and technical progress.

Modeling is a natural process for people, it is present in any activity. The introduction with nature by man occurs through constant modeling of situation, comparing with the basic models and past experience by them. Method for modeling, abstraction as a method of understanding the world is therefore an effective method of learning. Training activities associated with the creative transformation of the subject. The main feature of educational training activities is the systematic solution of the educational problems. The connection of the principles of developmental education, mathematical modeling, neurophysiological mechanisms of the brain and experience with Maple leads to the following conclusions: the method of mathematical modeling is not only scientific research but also the way of development of thinking in general; computer and mathematical environment (Maple), which is a powerful tool for scientific simulation can be considered as the elementary analogue of the brain. These qualities of computer mathematics led to the idea of using it not only as an effective methodological tool but as a means of nurturing the thinking and development of mental functions of the brain. To study this effect the school psychologist conducted a test, which confirm the observations: the dynamics of intellectual options students who working with Maple compares favorably with peers. In the process of doing computer math, in particular Maple, are involved in complex different mental functions. It is in the inclusion of all mental functions is the essence of integration of learning, its educational character. And this, in turn, contributes to the solution of moral problems.

Long-term work with computer mathematics led to the idea to use it as a tool for psychological testing. One of the projects focuses on the psychology and contains authors Maple–tests to identify the degree of development of different mental functions. Interactive mathematical environment gives a wide variability and creative testing capabilities. Moreover, Maple–test can be used not only as diagnostic but also as educational, and corrective. This technology was tested in one of psycho neurological dispensaries a few years ago.

Currently, one of the author's students, the so-called "homeworkers", the second year is a young man with a diagnosis, categories F20, who does not speak and does not write independently. It was impossible to get feedback from him and have basic training until then author have begun to apply computer-based tools, including system Maple. Working with the computer tests and mathematical objects helps to see not only the mental and even the simplest thinking movement, but also emotional movement!

In general, the effectiveness of the implementation in the structure of educational process of secondary school of new organizational forms of the use of computers, based on the application of the symbolic mathematics package Maple, computer modeling and information technology, has many aspects, here are some of them:

goals of education and math in particular;
additional education;
methodical and professional opportunities;
theoretical education;
modeling;
scientific creativity;
logical language;
spatial thinking, the development of the imagination;
programming skills;
the specificity of technical translation;
differentiation and individualization of educational process;
prospective teaching, the continuity of higher and secondary mathematics education;
development of creative abilities, research skills;
analytical thinking;
mathematical thinking;
mental diagnosis;
mental correction.

According to the author, the unique experience of the Kazan 57–th school suggests that computer mathematics (Maple) is the most effective also universal tool of new methods of inclusion. In recent decades, there are more children with a specific behavior, with a specific perception, not able to focus, with a poor memory, poor thinking processes. There are children, emotionally and intellectually healthy, or even ahead of their peers in one team together with them. High school should provide all the common core learning standards. It needed a variety of programs and techniques, as well as specialists who use them. Due to its remarkable features, computer mathematics, in particular Maple, can be used or be the basis of the variation of methods of physico-mathematical disciplines of inclusive education.

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